A tennis racket touches the ball for only a few thousandths of a second, yet it sends it flying. That brief burst of force is an impulse — and it’s exactly equal to the change in the ball’s momentum. Understanding impulse tells you why airbags save lives, why you bend your knees when landing, and why a cricketer draws their hands back to catch.
📘 What you need to know
Impulse is the product of force and the time it acts: J = FΔt (units N s)
Impulse equals the change in momentum:J = Δp = mv − mu
This follows from Newton’s second law written as force = rate of change of momentum
On a force–time graph, the impulse is the area under the curve
Impulse is a vector, pointing in the direction of the resultant force
A small force over a long time gives the same impulse as a large force over a short time
What impulse is
When a resultant force acts on an object for a short time and changes its motion — kicking a ball, catching it, a collision — we call that an impulse. It’s defined as the force multiplied by the time for which it acts:
ImpulseJ = FΔt
where J is the impulse in newton seconds (N s), F is the resultant force in newtons (N), and Δt is the time it acts in seconds (s). Because the force in a collision acts for such a tiny time, it’s very hard to measure directly — which is where the link to momentum becomes so useful.
Impulse equals change in momentum
Newton’s second law can be written as “the resultant force equals the rate of change of momentum”:
Newton’s second law (momentum form)F = Δp ÷ Δt ⇒ Δp = FΔt
The right-hand side is exactly the impulse. So impulse equals the change in momentum — a result you’ll use constantly:
Impulse–momentum theoremJ = Δp = mv − mu
Here m is the mass, v the final velocity, and u the initial velocity. This lets you find the impulse without measuring the force directly — you just need the object’s velocity before and after. (The equations apply when the force is constant.) It also means impulse and momentum share the same units: N s is the same as kg m s−1.
Watch the signs. Because impulse is a change in momentum, and momentum is a vector, a ball that reverses direction has a much bigger impulse than one that simply stops. Take rightward as positive: a ball hitting a wall at +25 and returning at −18 changes velocity by −43, not −7. Forgetting to flip the sign on the return velocity is the number-one impulse mistake in exams.
WE 1
A constant resultant force of 15 N acts on a trolley for 0.30 s. Calculate the impulse delivered to the trolley.
Step 1 — use the impulse definition
J = FΔt
Step 2 — substituteJ = 15 × 0.30J = 4.5 N sThat’s also the trolley’s change in momentum: 4.5 kg m s⁻¹.
Impulse on a force–time graph
Real collision forces aren’t constant — they rise to a peak and fall away in an instant. When the force varies, the impulse is the area under the force–time graph. This is one of the most reliable ways to find an impulse from experimental data.
When the force changes over time, the impulse is the shaded area under the force–time graph — which equals the total change in momentum.
WE 2
A 60 g ball moving to the right at 25 m s−1 is struck and returns to the left at 18 m s−1. Calculate the impulse delivered to the ball.
Step 1 — list values (right = positive)
m = 60 g = 0.060 kg
u = +25 m/s, v = −18 m/s (reversed!)
Step 2 — impulse = change in momentum
J = m(v − u)
J = 0.060 × (−18 − 25) = 0.060 × (−43)J = −2.6 N sThe minus sign means the impulse acts to the left — opposite the ball’s original motion. The reversal makes it much bigger than if the ball had just stopped.
Why the contact time matters
Rearrange the impulse–momentum theorem and something powerful appears: for a given change in momentum, the force depends on the time over which it acts.
Force from impulseF = Δp ÷ Δt
Stop something quickly and the force is huge; stretch the same change over a longer time and the force shrinks. This is the physics behind safety design everywhere — airbags, crumple zones, crash mats, and cushioned running shoes all increase the contact time to reduce the force on the body. A cricketer catching a fast ball does the same by hand: they draw their hands back as the ball arrives.
Same ball, same change in momentum — but drawing the hands back stretches the catch over more time, so the force on the hands is far smaller. That’s why it hurts less.
WE 3
A ball undergoes a change in momentum of 6.0 kg m s−1 when caught. Compare the average force on the hands if the catch takes 0.020 s (rigid hands) versus 0.20 s (hands drawn back).
Step 1 — use F = Δp ÷ ΔtStep 2 — rigid catch (0.020 s)F = 6.0 ÷ 0.020 = 300 NStep 3 — soft catch (0.20 s)F = 6.0 ÷ 0.20 = 30 N300 N vs 30 N — ten times smallerTen times the contact time means one-tenth the force. Same impulse, gentler landing.
💡 Top tips
Impulse = change in momentum. Use J = mv − mu when you can’t measure the force directly.
Mind the signs. A reversing object changes velocity by more than a stopping one — flip the sign of the return velocity.
Area under a force–time graph gives the impulse when the force isn’t constant.
Longer contact time = smaller force for the same impulse — the basis of all impact safety design.
N s and kg m s−1 are the same unit — impulse and momentum share it.
Quick recap: impulse is J = FΔt, and it equals the change in momentum, mv − mu. On a force–time graph it’s the area under the curve. For a fixed impulse, a longer contact time means a smaller force — the principle behind airbags, crumple zones, and a well-judged catch.
⚠ Common mistakes
Forgetting to reverse the sign of a rebounding object’s velocity — it changes momentum by more, not less
Using speed instead of velocity — impulse is a vector and needs directions
Treating a varying force as constant — use the area under the force–time graph instead
Confusing impulse (N s) with force (N) — impulse already includes the time
Thinking a longer contact time means a bigger force — it’s the opposite
Impulse is really Newton’s second law rewritten in terms of momentum — force as the rate of change of momentum. That momentum form deserves a closer look, especially where mass itself changes (like a rocket burning fuel). That’s next: Force & Momentum, and the equation F = Δp/Δt in full.
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